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Every Point on the Bisector of an Angle Is Equidistant from the Sides of the Angle.

If AD is me...

Question

If $A D$ is median of $△ A B C$ and $P$ is a point on $A C$ that $a r (A D P) : a r (A B D) = 2 : 3$ then $a r (P D C) : a r (A B C)$ is

1 : 5

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1 : 7

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1 : 6

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3 : 5

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Solution

The correct option is C $1 : 6$
$\frac{A r e a A D P}{A r e a A B D} = \frac{2}{3}$
Area $A D P = 2 x A r e a A B D = 3 x$
$A r . A B D = A r . A D C = 3 x$
$A r . A D C = A e . A D P + A r . A D C$
$\Rightarrow 3 x = 2 x + A r . P D C$
$A r . P D C = X$
Area. $A B C = A r . A D B = A r . A D C$
$= 3 x + 3 x = 6 x$
$\frac{A r . P D C}{A r . A B C} = \frac{x}{6 x}$
$= \frac{1}{6}$

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